2
$\begingroup$

I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE.

A bit of context: usually, Schilder's theorem tells us that the brownian motion with diffusivity $\nu>0$ (or rather, the Wiener measure $R^\nu$ on $\Omega=\{\omega\in C([0,1];\mathbb R^d)\}$) started from the origin satisfies a large deviation principle as $\nu\to 0$ with scale $\nu$ and rate function $A(\omega)=\frac 12\int_0^1 |\dot\omega_t|^2 dt$ (if $\omega$ is $H^1$ and $\omega_0=0$, and $A(\omega)=+\infty$ otherwise).

This however stands for the Brownian motion pinned at the origin (or at any other point, for that matter). What happens if one whishes to also pin the motion at the other endpoint $t=1$, in addition to $t=0$?

My specific question:

For fixed $x,y$ is there a Schilder theorem for the (law of the) Brownian bridge $R^{\nu,x,x}=R^\nu(\,\cdot\,|X_0=x,\,\cdot\,|X_1=y)$?

I suspect that the anser is yes, except that the rate function should be now $ A^{x,y}(\omega)=\frac 12 \int_0^1|\dot\omega_t|^2dt -\frac 12 |\omega_1-\omega_0|^2 $ if $\Big[\omega\in H^1$ with $\omega_0=x,\omega_1=y\Big]$, and $A^{x,y}(\omega)=+\infty$ otherwise. I am aware that this is ambiguous, since the conditionned $R^{\nu,x,y}$ is only defined for $R^{\nu}_{0,1}$-almost all $x,y$. (and in fact my real problem is set on the flat torus $\mathbb T^d$, and the Wiener measure I'm talking about is really the reversible one, i-e started from the stationary Lebesgue measure)

Note: I can only suspect that this is a possible way to recover the fact that the bridges $R^{\nu,x,y}$ converge to deterministic geodesics $[x\to y]$, since indeed $A^{x,y}$ is minimized by the unique geodesic $\omega^{x,y}_t=(1-t)x+ty$?

Strangely enough I could not find anything in this direction anywhere in the literature, including in the classical textbook of [Dembo & Zeitouni] on large deviations. Any help will be greatly appreciated.

$\endgroup$
4
  • $\begingroup$ I think that Corollary 4.9.3 of Bogachev's Gaussian Measures may be what you want. It certainly covers the case $x=y=0$. The rate function is going to come from the Cameron-Martin norm for Brownian bridge which I don't recall off the top of my head. For further references, see the bibliographic notes at the bottom of page 385. I will try to post more later if I have time. $\endgroup$ Sep 26, 2018 at 13:54
  • $\begingroup$ Thank you Nate for your time. I could get my hands on the 1998 version of Bogachev's book, but as far as I can see his corollary 4.9.3 deals with standard Brownian (centered Gaussian, i-e brownian started from $0$) and not with the bridges. But the book is quite dense so it's hard to navigate through all the notations and perhaps I missed a point? $\endgroup$ Sep 26, 2018 at 20:36
  • $\begingroup$ It's an arbitrary centered Gaussian measure, which includes the law of any continuous mean-zero Gaussian process. Brownian bridge which starts and ends at zero is certainly such a process. $\endgroup$ Sep 26, 2018 at 22:13
  • $\begingroup$ Aaaaaaah, I see. Thanks Nate, this clears things up. Again, sorry for my limited understanding of probabilistic notations/definitions, indeed centered Gaussian measures cover the case of my bridge $R^{\nu,x,y}$. $\endgroup$ Sep 27, 2018 at 6:31

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.